Optimal. Leaf size=34 \[ -2 \text {Li}_2\left (-e^x\right )-\frac {x^2}{e^x+1}+x^2-2 x \log \left (e^x+1\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2267, 6688, 2191, 2184, 2190, 2279, 2391} \[ -2 \text {PolyLog}\left (2,-e^x\right )-\frac {x^2}{e^x+1}+x^2-2 x \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2191
Rule 2267
Rule 2279
Rule 2391
Rule 6688
Rubi steps
\begin {align*} \int \frac {x^2}{2+e^{-x}+e^x} \, dx &=\int \frac {e^x x^2}{1+2 e^x+e^{2 x}} \, dx\\ &=\int \frac {e^x x^2}{\left (1+e^x\right )^2} \, dx\\ &=-\frac {x^2}{1+e^x}+2 \int \frac {x}{1+e^x} \, dx\\ &=x^2-\frac {x^2}{1+e^x}-2 \int \frac {e^x x}{1+e^x} \, dx\\ &=x^2-\frac {x^2}{1+e^x}-2 x \log \left (1+e^x\right )+2 \int \log \left (1+e^x\right ) \, dx\\ &=x^2-\frac {x^2}{1+e^x}-2 x \log \left (1+e^x\right )+2 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^x\right )\\ &=x^2-\frac {x^2}{1+e^x}-2 x \log \left (1+e^x\right )-2 \text {Li}_2\left (-e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 33, normalized size = 0.97 \[ x \left (\frac {e^x x}{e^x+1}-2 \log \left (e^x+1\right )\right )-2 \text {Li}_2\left (-e^x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 38, normalized size = 1.12 \[ \frac {x^{2} e^{x} - 2 \, {\left (e^{x} + 1\right )} {\rm Li}_2\left (-e^{x}\right ) - 2 \, {\left (x e^{x} + x\right )} \log \left (e^{x} + 1\right )}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{e^{\left (-x\right )} + e^{x} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 32, normalized size = 0.94 \[ x^{2}-\frac {x^{2}}{{\mathrm e}^{x}+1}-2 x \ln \left ({\mathrm e}^{x}+1\right )-2 \polylog \left (2, -{\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 30, normalized size = 0.88 \[ x^{2} - 2 \, x \log \left (e^{x} + 1\right ) - \frac {x^{2}}{e^{x} + 1} - 2 \, {\rm Li}_2\left (-e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2}{{\mathrm {e}}^{-x}+{\mathrm {e}}^x+2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {x^{2}}{e^{x} + 1} + 2 \int \frac {x}{e^{x} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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